# Connection Cochain of Abelian Extensions and Connection $1$-Forms

## TL;DR

This paper explores the relationship between connection cochains of abelian extensions and connection 1-forms on principal bundles, clarifying their interplay in the context of abelian groups.

## Contribution

It extends the concept of connection cochains to abelian extensions and relates them explicitly to connection 1-forms in principal bundles.

## Key findings

- Connection cochains of abelian extensions are shown to correspond to connection 1-forms.
- The paper clarifies the relationship between algebraic and geometric notions of connections.
- Provides a framework for understanding connections in abelian principal bundles.

## Abstract

In this paper, we consider the concept of connection cochain of central extensions introduced by Moriyoshi and apply it to the abelian case. We will show the relationship between connection cochain and connection $1$-form of a principal bundle whose structure group is abelian.